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Zbl 0834.62068
Pukelsheim, Friedrich
Optimal design of experiments. (English)
[B] New York, NY: Wiley. xxiii, 454 p. (1993). ISBN 0-471-61971-X

The aim of the book is to present the ``Optimality theory of experimental designs in linear models''. ``The design problems originate from statistics, but are solved using special tools from linear algebra and convex analysis'' (all from the preface). Consequently, a matrix oriented approach is chosen which uses information functionals rather than functionals of the covariance matrices, and the optimization problems are described in terms of subgradients and polars. By its high theoretical level this textbook is complementary to the series of so far existing monographs on optimal designs, e.g. by {\it V. V. Fedorov} [Theory of optimal experiments. (1972); translation of the original Russian edition (1971; Zbl 0261.62002)], {\it H. Bandemer} [Theorie und Anwendung der optimalen Versuchsplanung. I: (1977; Zbl 0391.62001); II: (1980; Zbl 0486.62067)], {\it S. D. Silvey} [Optimal design. (1980; Zbl 0468.62070)], {\it A. Pázman} [Foundations of optimum experimental design. (1986; Zbl 0588.62117)]\ or {\it A. C. Atkinson} and {\it A. N. Donev} [Optimum experimental designs. (1992; Zbl 0829.62070)]. \par The present book is a comprehensive source of background information on the optimality of designs rather than a collection of recipes for the construction. Its material is arranged into fifteen sections including a matrix oriented summary of linear models, a problem description illustrated by one-dimensional optimization tasks emphasizing the importance of the Elfving set, optimality criteria based on the information matrices like uniform (Loewner) optimality and matrix means, optimality results centered around the corresponding general equivalence theorems, and further topics like admissibility, Bayesian designs, efficient approximations for finite sample sizes and invariance considerations leading to the concept of universal (Kiefer) optimality. \par Applications are mainly devoted to polynomial regression models. The text is augmented by comments on the literature, biographical notes on Loewner, Elfving and Kiefer, an extensive bibliography and an exhaustive subject index.
[R.Schwabe (Berlin)]

MSC 2000:: *62K05 Optimal statistical designs
62-02 Research monographs (statistics)
62-01 Textbooks (statistics)

Keywords: convex optimization; universal optimality; matrix oriented approach; information functionals; subgradients; polars; linear models; Elfving set; general equivalence theorems; admissibility; Bayesian designs; efficient approximations for finite sample sizes; invariance considerations; polynomial regression models; bibliography

Citations: Zbl 0261.62002; Zbl 0391.62001; Zbl 0486.62067; Zbl 0468.62070; Zbl 0588.62117; Zbl 0829.62070

Cited in: Zbl 1106.62078 Zbl 1101.62063 Zbl 0898.62094 Zbl 0874.62080 Zbl 0910.62068 Zbl 0835.62062

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